In synchronous digital data communication systems, the carrier phase and symbol timing of the received signal must be acquired and tracked by the receiver in order to respectively demodulate the received signal and to recover the transmitted data from the received signal. Typically, receivers require carrier phase tracking for signal demodulation and symbol time tracking for data detection for generating received data streams.
Continuous phase modulation (CPM) provides a class of digital phase modulation signals that have a constant envelope. The spectral occupancy of a CPM signal can be controlled or tailored to the available bandwidth of a transmission channel. The constant envelope CPM signals allow saturated power amplifier operation for maximum power efficiency. The use of CPM signals in communications systems can potentially achieve significant improvement in both power and spectral efficiency over other conventional modulation techniques, at the cost of a moderate increase in receiver complexity. Bit error rate reduction has been achieved using trellis CPM demodulation with ideal synchronization. There is a continuing need to develop hardware implementation of the symbol time and carrier phase synchronizers that provides required tracking functions for the coherent CPM receiver. Often, symbol time tracking and carrier phase tracking limit the performance of CPM systems.
A particular type of CPM system is a Gaussian minimum shift keying (GMSK) system where a data sequence is precoded and the precoded data symbols are used for continuous phase modulation. The GMSK received signals are filtered using Laurent filters and samplers for providing data samples subject to trellis demodulation for generating an estimate of the data sequence. Carrier phase tracking loops are used for demodulating the received signal by tracking the carrier phase, and symbol time tracking loops are used for synchronized sampling of Laurent matched filter signals for generating the data samples that used to generate estimates of the transmitted bit stream using trellis demodulation. These carrier phase and symbol time tracking loops are often referred to as synchronizer. These synchronizers often lose track during noisy communications.
A binary continuous phase modulation signal can be described by complex envelop equations.
            z      ⁡              (        t        )              =          Re      ⁡              (                                            z              b                        ⁡                          (              t              )                                ⁢                      ⅇ                          j2π              ⁢                                                          ⁢                              f                c                            ⁢              t                                      )                                z        b            ⁡              (        t        )              =                            2          ⁢                                    E              b                        /            T                              ⁢              ⅇ                  j          ⁢                                          ⁢                      ϕ            ⁡                          (                              t                ,                α                            )                                                                                    ϕ            ⁡                          (                              t                ,                α                            )                                =                      π            ⁢                                                  ⁢            h            ⁢                                          ∫                                  -                  ∞                                t                            ⁢                                                ∑                                      n                    =                    0                                                        N                    -                    1                                                  ⁢                                                                  ⁢                                                      α                    n                                    ⁢                                      f                    ⁡                                          (                                              t                        -                        nT                                            )                                                        ⁢                                                                          ⁢                                      ⅆ                    t                                                                                                                                =                          π              ⁢                                                          ⁢              h              ⁢                                                ∑                                      n                    =                    0                                                        N                    -                    1                                                  ⁢                                                                  ⁢                                                      α                    n                                    ⁢                                      g                    ⁡                                          (                                              t                        -                        nT                                            )                                                                                                    ⁢                                                    
The term Zb(t) is called the complex envelope of the CPM signal, fc is the carrier frequency, Eb is the bit energy, T is the bit duration, and N is the transmitted data length in bits, α=(α0α1, . . . αN−1,)αi∈{±1}, represents one of 2N equally probable data sequences. The parameter h is the modulation index, f(t) is the pulse response of the smoothing filter in the CPM modulator, and g(t) is the CPM phase response defined in terms of the f(t) pulse response.
      g    ⁡          (      t      )        ⁢          =            ∫              -        ∞            t        ⁢                  ⁢                  f        ⁡                  (          s          )                    ⁢                          ⁢              ⅆ        s            
The pulse response f(t) is limited to the time interval [0,LT] for some integer L and having the properties that f(t)=f(LT−t) and g(Lf)=1. The pulse amplitude modulation (PAM) representation of signal CPM envelope is well known. Laurent has shown that the complex envelope zb(t) can be expressed as a double summation.
            z      b        ⁡          (      t      )        =                    2        ⁢                              E            b                    /          T                      ⁢                  ∑                  k          =          0                          2                      L            -            1                              ⁢                          ⁢                        ∑                      n            =            0                                N            -            1                          ⁢                              α                          k              ,              n                                ⁢                                    h              k                        ⁡                          (                              t                -                kT                            )                                          
In this PAM representation of the baseband CPM signal envelope, also referred to as the Laurent decomposition, the ak,n values are known as pseudo data symbols and are related to the modulated data symbols generally by a pseudo data symbol equation.
      a          k      ,      n        =      exp    ⁡          (              jh        ⁢                                  ⁢                  π          ⁡                      [                                                            ∑                                      m                    =                    0                                    n                                ⁢                                  α                  m                                            -                                                ∑                                      i                    =                    0                                                        L                    -                    1                                                  ⁢                                                      α                                          n                      -                      i                                                        ⁢                                      β                                          k                      ,                      i                                                                                            ]                              )      
In the pseudo data symbol equation, for all k, 0≦k≦2L−1, βk,0=0 and βki is a 0 or 1 digit in the binary expansion of
  k  =            ∑              i        =        1                    L        -        1              ⁢                  2                  i          -          1                    ⁢              2                  i          -          1                    ⁢                        β                      k            ,            i                          .            These pseudo data symbols take on values in the set {±1, ±j} when the modulation index h equals ½. In general, the first two pseudo data symbols, a0,n and a1,n can be written in an expanded form.
            a              0        ,        n              =                  exp        ⁡                  (                      j            ⁢                                                  ⁢            πh            ⁢                                          ∑                                  m                  =                  0                                n                            ⁢                              α                m                                              )                    =                        a                      0            ,                          n              -              1                                      ⁢                  J                      α            n                                ,          ⁢            a              0        ,                  -          1                      =    1    ,      J    =                            e          jxh                ⁢                  a                      1            ,            n                              =                        a                      0            ,                          n              -              L                                      ⁢                  J                      α            n                          ⁢                  J                      α                          n              -              2                                      ⁢                  J                      α                          n              -              3                                      ⁢                                  ⁢        …        ⁢                                  ⁢                  J                      α                          n              -              L              +              1                                          
The set of pulse functions {hk(t)}, termed Laurent pulse functions, have a real value and are finite in duration, and are formed by an hk(t) equation.
                    h        k            ⁡              (        t        )              =                  ∑                  i          =          0                          L          -          1                    ⁢              c        ⁡                  (                      t            +            iT            +                                          (                                                      β                                          k                      ,                      i                                                        -                  1                                )                            ⁢              LT                                )                      where            c      ⁡              (        t        )              =          (                                                                                    sin                  ⁡                                      (                                                                  π                        ⁢                                                                                                  ⁢                        h                                            -                                              π                        ⁢                                                                                                  ⁢                                                  hg                          ⁡                                                      (                                                                                        t                                                                                      )                                                                                                                )                                                  /                                  sin                  ⁡                                      (                                          π                      ⁢                                                                                          ⁢                      h                                        )                                                              ,                                                                                          t                                            ≤              LT                                                                          0              ,                                            elsewhere                              )      
Among these hk(t) pulses, most of the signal energy is carried by the principal Laurent pulse h0(t), which has a duration of L+1 bit times. Another property of the principal Laurent pulse h0(t) is that it is symmetrical about t=(L+1) T/2. The principal Laurent function h0(t) output provides a gross estimate of the transmitted symbol sequence. These properties of the principal Laurent pulse function h0(t) have not yet been exploited in developing the error signals for the symbol time and carrier phase tracking loops. These and other disadvantages are solved or reduced using the invention.